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Q. Suppose that $g(x)$ is a twice differentiable function and $g(1) = 1, g(2) = 4$ and $g(3) = 9$.

Which of the following is necessarily true?

(a) $g''(x) = 2$ for some $x \in [1,3]$

(b) $g''(x) = 2$ for some $x \in [1.5,2.5]$

I did this by applying MVT twice and found 1st option to be correct.

Please help If I am making some mistake or wrong assumptions?

Shubham Singh
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    Welcome to MSE. Please use MathJax. – José Carlos Santos May 21 '18 at 14:37
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    $(a)$ is true. In general, if $f(x)$ is defined on $[a, a+nh]$, continuously differentiable up to $n$ times, then there is a $c \in (a,a+nh)$ such that $f^{(n)}(c)$ equals to $n$-th finite difference of $f$ at $a$. More precisely, $f^{(n)}(c) = \frac{1}{n!}\Delta_h^n f(x)|{x=a} \stackrel{def} = \sum{k=0}^n (-1)^{n-k}\binom{n}{k}f(a+kh)$. For a proof, see this answer. – achille hui May 21 '18 at 14:45
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    If these were the only two options on a multiple-choice test, then you can note that (b) implies (a), so if only one choice is true, then it must be (a). – Connor Harris May 21 '18 at 16:39
  • @ConnorHarris No there were two more options but I was confused between these two only. – Shubham Singh May 21 '18 at 17:11

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